scholarly journals Deformation quantization on the cotangent bundle of a Lie group

2021 ◽  
Vol 62 (3) ◽  
pp. 033504
Author(s):  
Ziemowit Domański
Keyword(s):  
Lie Group ◽  
Deformation Quantization ◽  
Cotangent Bundle

The Duflo non-commutative Fourier transform for the Lorentz group

2020 ◽  
Vol 17 (supp01) ◽  
pp. 2040011
Author(s):  
Giacomo Rosati
Keyword(s):  
Fourier Transform ◽  
Phase Space ◽  
Quantum System ◽  
Lie Group ◽  
Lorentz Group ◽  
Commutative Algebra ◽  
Cotangent Bundle ◽  
Irreducible Representations ◽  
Algebra Representation

For a quantum system whose phase space is the cotangent bundle of a Lie group, like for systems endowed with particular cases of curved geometry, one usually resorts to a description in terms of the irreducible representations of the Lie group, where the role of (non-commutative) phase space variables remains obscure. However, a non-commutative Fourier transform can be defined, intertwining the group and (non-commutative) algebra representation, depending on the specific quantization map. We discuss the construction of the non-commutative Fourier transform and the non-commutative algebra representation, via the Duflo quantization map, for a system whose phase space is the cotangent bundle of the Lorentz group.

scholarly journals On the BKS pairing for Kähler quantizations of the cotangent bundle of a Lie group

2006 ◽  
Vol 234 (1) ◽  
pp. 180-198 ◽  
Author(s):  
Carlos Florentino ◽  
Pedro Matias ◽  
José Mourão ◽  
João P. Nunes
Keyword(s):  
Lie Group ◽  
Cotangent Bundle

scholarly journals Hyperkähler metrics associated to compact Lie groups

1996 ◽  
Vol 120 (1) ◽  
pp. 61-69 ◽  
Author(s):  
Andrew Dancer ◽  
Andrew Swann
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Cotangent Bundle ◽  
Symplectic Structure ◽  
Compact Lie Group ◽  
Compact Lie Groups ◽  
Left And Right ◽  
Hyperkähler Metrics

It is well known that the cotangent bundle of any manifold has a canonical symplectic structure. If we specialize to the case when the manifold is a compact Lie group G, then this structure is preserved by the actions of G on T*G induced by left and right translation on G. We refer to these as the left and right actions of G on T*G.

scholarly journals Poisson structures on the cotangent bundle of a Lie group or a principle bundle and their reductions

1994 ◽  
Vol 35 (9) ◽  
pp. 4909-4927 ◽  
Author(s):  
D. Alekseevsky ◽  
J. Grabowski ◽  
G. Marmo ◽  
P. W. Michor
Keyword(s):  
Lie Group ◽  
Cotangent Bundle ◽  
Poisson Structures

HOLOMORPHIC SYMPLECTIC AND POISSON STRUCTURES ON THE HOLOMORPHIC COTANGENT BUNDLE OF A COMPLEX LIE GROUP AND OF A HOLOMORPHIC PRINCIPAL BUNDLE

2013 ◽  
Vol 06 (03) ◽  
pp. 1350029 ◽  
Author(s):  
Cristian Ida
Keyword(s):  
Lie Group ◽  
Cotangent Bundle ◽  
Principal Bundle ◽  
Poisson Structures

In this paper we introduce holomorphic symplectic and Poisson structures on the holomorphic cotangent bundle of a complex Lie group and of a holomorphic principal bundle.

scholarly journals GAUGE-INVARIANT HAMILTONIAN FORMULATION OF LATTICE YANG-MILLS THEORY AND THE HEISENBERG DOUBLE

1995 ◽  
Vol 10 (34) ◽  
pp. 2619-2631 ◽  
Author(s):  
S.A. FROLOV
Keyword(s):  
Lie Group ◽  
Cotangent Bundle ◽  
Hamiltonian Formulation ◽  
Dimensional Torus ◽  
Gauge Invariant ◽  
Flat Connections ◽  
Yang Mills ◽  
Temporal Gauge ◽  
Heisenberg Double ◽  
Limiting Case

It is known that to get the usual Hamiltonian formulation of lattice Yang-Mills theory in the temporal gauge A0=0 one should place on each link a cotangent bundle of a Lie group. The cotangent bundle may be considered as a limiting case of a so-called Heisenberg double of a Lie group which is one of the basic objects in the theory of Lie-Poisson and quantum groups. It is shown in the paper that there is a generalization of the usual Hamiltonian formulation to the case of the Heisenberg double. The physical phase space of the (1+1)-dimensional γ-deformed Yang-Mills model is proved to be equivalent to the moduli space of flat connections on a two-dimensional torus.

scholarly journals Bi-Hamiltonian Structure of Spin Sutherland Models: The Holomorphic Case

2021 ◽  
Author(s):  
L. Fehér
Keyword(s):  
Lie Group ◽  
Poisson Structure ◽  
Cotangent Bundle ◽  
Symplectic Structure ◽  
Hamiltonian Structure ◽  
Hamiltonian Structures ◽  
Poisson Reduction ◽  
Sutherland Model ◽  
Heisenberg Double ◽  
Real Forms

AbstractWe construct a bi-Hamiltonian structure for the holomorphic spin Sutherland hierarchy based on collective spin variables. The construction relies on Poisson reduction of a bi-Hamiltonian structure on the holomorphic cotangent bundle of $$\mathrm{GL}(n,\mathbb {C})$$ GL ( n , C ) , which itself arises from the canonical symplectic structure and the Poisson structure of the Heisenberg double of the standard $$\mathrm{GL}(n,\mathbb {C})$$ GL ( n , C ) Poisson–Lie group. The previously obtained bi-Hamiltonian structures of the hyperbolic and trigonometric real forms are recovered on real slices of the holomorphic spin Sutherland model.

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